It suffices to prove the statement for the \(\overline{\partial }\) –Neumann operator ([1]}; [2]}, Theorem 5.5). The proof follows the first part of the proof of Theorem 1 in section 4 of [3]}, but then uses the commutator formulas from [4]} (i.e.Lemmas REF  and REF  above) instead of Lemmas 4 and 5 in [3]}.
We use a downward induction on the degree \(q\) . In the top degree \(q=n\) , the \(\overline{\partial }\) –Neumann boundary conditions reduce to Dirichlet boundary conditions, and \(N_{n}\)  gains two derivatives in Sobolev norms. So to show that \(N_{q_{0}}\)  satisfies Sobolev estimates, it suffices to show: if \(N_{q+1}\)  satisfies Sobolev estimates, and \(q\ge q_{0}\) , then so does \(N_{q}\) . So fix such a \(q\) . The induction assumption will be invoked to conclude that the Bergman projection \(P_{q}\)  on \((0,q)\) –forms satisfies Sobolev estimates ([2]}, Theorem 5.5).
As usual, the arguments will involve absorbing terms, and so one has to know that these terms are finite. In order to insure that, we work with the regularized operators \(N_{\delta ,q}\)  that result form elliptic regularization. That is, \(N_{\delta ,q}\)  is the inverse of the selfadjoint operator \(\Box _{\delta ,q}\)  associated with the quadratic form \(Q_{\delta ,q}(u,u)=\Vert \overline{\partial }u\Vert ^{2}+\Vert \overline{\partial }^{*}u\Vert ^{2}+\delta \Vert \nabla u\Vert ^{2}\) , with form domain \(W^{1}_{(0,q)}(\Omega )\cap dom(\overline{\partial }^{*})\) , where \(\nabla \)  denotes the vector of all first order derivatives of all coefficients of \(u\) . \(N_{\delta ,q}\)  maps \(L^{2}_{(0,q)}(\Omega )\)  continuously into the form domain, endowed with the norm \(Q_{\delta ,q}(u,u)^{1/2}\) . We will prove estimates with constants that are uniform in \(\delta \) ; letting \(\delta \rightarrow 0\)  then gives the desired estimates for \(N_{q}\)  ([7]}, pages 102–103 and [2]}, section 3.3).
So \(q\)  is now fixed, \(q_{0}\le q\) , and \(N_{q+1}\) , hence \(P_{q}\) , satisfy Sobolev estimates. We want to prove Sobolev estimates for \(N_{\delta ,q}\) , with constants uniform in \(\delta \)  (\(\delta \)  small). Because \(P_{q}\)  is regular in Sobolev norms, Lemma 7 in [3]} says that for all \(s>0\) , there is a constant \(C_{s}\)  independent of \(\delta \)  such that we have the estimate
\(\Vert N_{\delta ,q}u\Vert _{s} \le C_{s}(\Vert \overline{\partial }N_{\delta ,q}u\Vert _{s} + \Vert \overline{\partial }^{*}N_{\delta ,q}u\Vert _{s})\;.\) 
Therefore, it suffices to prove estimates for the right hand side of (REF ) (with constants independent of \(\delta \) ). In order to bring \(Q_{\delta ,q}\)  into play, we prove estimates for \(\overline{\partial }N_{\delta ,q}\) , \(\overline{\partial }^{*}N_{\delta ,q}\) , and \(\delta ^{1/2}\nabla N_{\delta ,q}\)  as in [3]}. We do this by induction on the Sobolev index \(k\) .
Because \(N_{\delta ,q}\)  maps \(L^{2}_{(0,q)}(\Omega )\)  continuously into the form domain of \(Q_{\delta ,q}\) , with norm less than or equal to 1, the following inductive assumption holds for \(l=0\) : there are \(\eta \in (0,1)\) , a constant \(C\)  and \(\delta _{0}>0\)  such that
\(\Vert \overline{\partial }N_{\delta ,q}u\Vert _{l,2h_{\eta }}^{2}+\Vert \overline{\partial }^{*}N_{\delta ,q}u\Vert _{l,2h_{\eta }}^{2}+\delta \Vert \nabla N_{\delta ,q}u\Vert _{l,2h_{\eta }}^{2} \le C\Vert u\Vert _{l}^{2}\;,\;0<\delta \le \delta _{0}\;.\) 
Indeed, take any \(\eta \in (0,1)\) , then choose \(C\)  big enough and \(\delta >0\)  arbitrary. We now assume this induction assumption holds for \(0\le l\le (k-1)\)  and show that it then holds for \(l=k\) . The induction assumption (REF ) corresponds to (31) in [3]}, with the modification that the norms on the left hand side are weighted, and that there is no term \(\delta ^{2}\Vert u\Vert _{l+1}^{2}\) . It turns out that with Lemmas REF  and REF  for the commutators with \(\overline{\partial }\)  and \(\overline{\partial }^{*}\)  (rather than Lemmas 4 and 5 from [3]}), this term is not needed for the induction to run.
For \(\eta \in (0,1)\) , set \(X_{\eta }=e^{-h_{\eta }+\rho g_{\eta }}L_{n}^{\rho }\) , where \(g_{\eta }\)  is a smooth function that agrees on the boundary with \(\alpha _{\eta }(\overline{L_{n}})/\overline{L_{n}}\rho \) . The reason for this choice of exponent (rather than just \(-h_{\eta }\) ) is that the terms where \(j=n\)  in the commutator (REF ) below will then vanish on the boundary. This fact is needed in the argument. Being able to make this choice is a manifestation of the principle that `commutator conditions in the normal (and any strictly pseudoconvex) direction come for free' ([2]}, Section 5.7, [14]}, proof of the lemma). Note that because \(g_{\eta }\)  is only prescribed on the boundary, we may assume that \(|\rho g_{\eta }|\le 1/2k\) . As usual, we need tangential differential operators to preserve the domain of \(\overline{\partial }^{*}\) , and so we let them act in special boundary charts ([2]}, section 2.2). The error this introduces is one order lower than the operator, and as a result, our estimates are not affected.
Using Lemma REF  for barred derivatives or derivatives that are complex tangential, we have
\(\Vert \overline{\partial }^{*}N_{\delta ,q}u\Vert _{k,2h_{\eta }}^{2} \lesssim \Vert (e^{-h_{\eta }}(L_{n}^{\rho }-\overline{L_{n}^{\rho }}))^{k}\overline{\partial }^{*}N_{\delta ,q}u\Vert ^{2} \\+C_{\eta }(\Vert \overline{\partial }\overline{\partial }^{*}N_{\delta ,q}u\Vert _{k-1}^{2}+\Vert \overline{\partial }^{*}N_{\delta ,q}u\Vert _{k-1}\Vert \overline{\partial }^{*}N_{\delta ,q}u\Vert _{k}+\Vert u\Vert _{k}^{2})\;.\) 
The \(\Vert u\Vert _{k}^{2}\)  term is needed because of the compactly supported term from the partition of unity used for the special boundary charts; this term is not covered by Lemma REF . The estimate follows from interior elliptic regularity of \(\Box _{\delta ,q}\) , with constants that are uniform in \(\delta \) . This uniformity holds because for \(u\in dom(\Box _{\delta ,q})\) , \(\Box _{\delta ,q}u=-(1/4+\delta )\Delta u\) , where \(\Delta \)  acts coefficientwise (see [2]}, formula (3.22)). We have used that when there is a barred derivative, or a complex tangential derivative, we can always commute it so that it acts first. Then there is no issue with with the requirement that a form must be in the domain of \(\overline{\partial }^{*}\)  in order for Lemma REF  to apply. The error term this makes is of order at most \((k-1)\) . Modulo terms controlled by \(\Vert \overline{\partial }^{*}N_{\delta ,q}u\Vert _{k}^{2}\) , \(\Vert (e^{-h_{\eta }}(L_{n}^{\rho }-\overline{L_{n}^{\rho }}))^{k}\overline{\partial }^{*}N_{\delta ,q}u\Vert ^{2} \lesssim \Vert e^{-kh_{\eta }}(L_{n}^{\rho }-\overline{L_{n}^{\rho }})^{k}\overline{\partial }^{*}N_{\delta ,q}u\Vert ^{2}\le e\Vert (X_{\eta }-\overline{X_{\eta }})^{k}\overline{\partial }^{*}N_{\delta ,q}u\Vert ^{2}\)  (since \(2k|\rho g_{\eta }|\le 1\) ). Upon also estimating the last term in (REF ) via a s.c.–l.c argument and absorbing the term \(s.c\Vert \overline{\partial }^{*}N_{\delta ,q}u\Vert _{k,2h_{\eta }}^{2}\)  into the left hand side of (REF ), we thus arrive at
\(\Vert \overline{\partial }^{*}N_{\delta ,q}u\Vert _{k,2h_{\eta }}^{2} \lesssim \Vert (X_{\eta }-\overline{X_{\eta }})^{k}\overline{\partial }^{*}N_{\delta ,q}u\Vert ^{2}+C_{\eta }(\Vert \overline{\partial }\overline{\partial }^{*}N_{\delta ,q}u\Vert _{k-1}^{2}+\Vert \overline{\partial }^{*}N_{\delta ,q}u\Vert _{k-1}^{2}+\Vert u\Vert _{k}^{2})\;.\) 
This estimate corresponds to estimate (32) in [3]}, with \(\eta \)  in the role of \(\varepsilon \) . Note that uniform boundedness of \(h_{\eta }\) , used in [3]} for estimate (32), is not needed here because we use the weighted norm on the left hand side.
Applying the same reasoning to \(\Vert \overline{\partial }N_{\delta ,q}u\Vert _{k,2h_{\eta }}^{2}\)  takes a little more care, because \(\overline{\partial }N_{\delta ,q}u\)  need not be in the domain of \(\overline{\partial }^{*}\)  (the free boundary condition imposed by \(Q_{\delta }\)  is different from that imposed by \(Q\) ). The necessary modification is explained in detail in [3]}, to which we refer the reader. The result is the following estimate:
\(\Vert \overline{\partial }N_{\delta ,q}u\Vert _{k,2h_{\eta }}^{2} \lesssim \Vert (X_{\eta }-\overline{X_{\eta }})^{k}\overline{\partial }N_{\delta ,q}u\Vert ^{2} \\+ C_{\eta }(\Vert \vartheta \overline{\partial }N_{\delta ,q}u\Vert _{k-1}^{2}+\Vert \overline{\partial }N_{\delta ,q}u\Vert _{k-1}^{2}+\Vert \delta (\partial /\partial \nu )N_{\delta ,q}u\wedge \overline{\omega _{n}}\Vert _{k}^{2}+\Vert u\Vert _{k}^{2})\;.\) 
Here, \(\vartheta \)  is the formal adjoint of \(\overline{\partial }\) , \((\partial /\partial \nu )\)  denotes the normal derivative acting coefficientwise on forms, and \(\omega _{n}\)  is the form dual to the unit normal \(L_{n}\) .
The argument for \(\delta \Vert \nabla N_{\delta ,q}u\Vert _{k,2h_{\eta }}^{2}\)  is slightly different from that given in [3]} (the argument there uses uniform bounds on the functions \(h_{\eta }\) ). At issue is the \(\partial /\partial \nu \)  term in \(\nabla \) , which need not be in the domain of \(\overline{\partial }^{*}\) . However, because the boundary is not characteristic for \(\overline{\partial }\oplus \vartheta \)  ([2]}, Lemma 2.2), we can write \(\nabla N_{\delta ,q}u = \nabla _{T}N_{\delta ,q}u\)  plus a linear combination of coefficients of \(\overline{\partial }N_{\delta ,q}u\) , \(\overline{\partial }^{*}N_{\delta ,q}u\) , and \(N_{\delta ,q}u\)  itself. Here, \(\nabla _{T}\)  stands for tangential derivatives. Applying the same reasoning as for (REF ) to \(\nabla _{T}N_{\delta ,q}u\)  gives
\(\delta \Vert \nabla N_{\delta ,q}u\Vert _{k,2h_{\eta }}^{2} \lesssim \delta \Vert (X_{\eta }-\overline{X_{\eta }})^{k}\nabla N_{\delta ,q}u\Vert ^{2} \\+ C_{\eta }\delta (\Vert \overline{\partial }N_{\delta ,q}u\Vert _{k}^{2}+\Vert \overline{\partial }^{*}N_{\delta ,q}u\Vert _{k}^{2} + \Vert N_{\delta ,q}u\Vert _{k}\;\Vert N_{\delta ,q}u\Vert _{k+1})\;.\) 
We have used that \(\Vert (X_{\eta }-\overline{X_{\eta }})^{k}\nabla _{T} N_{\delta ,q}u\Vert ^{2} \le \Vert (X_{\eta }-\overline{X_{\eta }})^{k}\nabla N_{\delta ,q}u\Vert ^{2}\) .
Estimates (REF ),(REF ), and (REF ) correspond to estimates (32), (33), and (34) in [3]}. It is shown there in (31)–(39), and the paragraph immediately following (39), that completing the induction step reduces to estimating the following three inner products:
\(\left((X_{\eta }-\overline{X_{\eta }})^{k}\overline{\partial }N_{\delta ,q}u,[\overline{\partial },X_{\eta }-\overline{X_{\eta }}](X_{\eta }-\overline{X_{\eta }})^{k-1}N_{\delta ,q}u\right)\;,\) 
\(\left((X_{\eta }-\overline{X_{\eta }})^{k}\overline{\partial }^{*}N_{\delta ,q}u,[\overline{\partial }^{*},X_{\eta }-\overline{X_{\eta }}](X_{\eta }-\overline{X_{\eta }})^{k-1}N_{\delta ,q}u\right)\;,\) 
and
\(\delta \left((X_{\eta }-\overline{X_{\eta }})^{k}\nabla N_{\delta ,q}u,[\nabla ,X_{\eta }-\overline{X_{\eta }}](X_{\eta }-\overline{X_{\eta }})^{k-1}N_{\delta ,q}u\right)\;.\) 
More precisely, it is shown in [3]} that for \(\eta \)  given, there is \(\delta _{0}(\eta )\)  such that all the error terms in (REF )–(REF ) are acceptable for (REF ) (for \(l=k\) ), or can be absorbed into the left hand side of (REF ) when \(\delta \le \delta _{0}(\eta )\) . Note that upon adding (REF ) through (REF ), using Cauchy–Schwarz on each inner product, followed by a s.c.–l.c. argument, the squares of the norms of the left hand sides in (REF )–(REF ) can be absorbed into the first term on the right hand side of the sum of (REF )–(REF ). Therefore, in order to estimate the sum of the left hand sides of (REF )–(REF ), we only have to estimate
\(\Vert [\overline{\partial },X_{\eta }-\overline{X_{\eta }}](X_{\eta }-\overline{X_{\eta }})^{k-1}N_{\delta ,q}u\Vert ^{2}+\Vert [\overline{\partial }^{*},X_{\eta }-\overline{X_{\eta }}](X_{\eta }-\overline{X_{\eta }})^{k-1}N_{\delta ,q}u\Vert ^{2} \\+ \delta \Vert [\nabla ,X_{\eta }-\overline{X_{\eta }}](X_{\eta }-\overline{X_{\eta }})^{k-1}N_{\delta ,q}u\Vert ^{2}\;.\) 
This expression differs from
\(\Vert [\overline{\partial },(X_{\eta }-\overline{X_{\eta }})^{k}]N_{\delta ,q}u\Vert ^{2}+\Vert [\overline{\partial }^{*},(X_{\eta }-\overline{X_{\eta }})^{k}]N_{\delta ,q}u\Vert ^{2}+ \delta \Vert [\nabla ,(X_{\eta }-\overline{X_{\eta }})^{k}]N_{\delta ,q}u\Vert ^{2}\) 
by terms that are of order \((k-1)\) . This follows from the formula for commutators with powers of an operator in [24]}, Lemma 2, page 418 (see also [3]}, formula (3.54)): \([A,T^{k}] = \sum _{j=1}^{k}\binom{k}{j}[\cdots [[A,T],T]\cdots ]T^{k-j}\)  (\(j\) –fold, note that these iterated commutators are of order one). In view of (REF ) and the induction assumption, these terms are acceptable. We will now estimate (REF ).
For the commutators with \(\overline{\partial }\)  and \(\overline{\partial }^{*}\)  we use Lemmas REF  and REF , with \(\psi = h_{\eta }-\rho g_{\eta }\)  and \(v=(X_{\eta }-\overline{X_{\eta }})^{k-1}N_{\delta ,q}u\) . Observe that near the boundary
\(\overline{L_{j}}\psi +\alpha ^{\rho }(\overline{L_{j}}) = -\overline{L_{j}}(\rho g_{\eta })+\overline{L_{j}}h_{\eta }+\alpha ^{\rho }(\overline{L_{j}}) = -\overline{L_{j}}(\rho g_{\eta }) + \alpha _{\eta }(\overline{L_{j}})\;; \;j<n\;;\) 
the last equality follows from (REF ). Inserting this equality into Lemma REF  and estimating the error terms with Lemma REF  gives
\(\Vert [\overline{\partial },(X_{\eta }-\overline{X_{\eta }})^{k}]N_{\delta ,q}u\Vert ^{2} \\\lesssim \mathop {\mmlmultiscripts{\sum {\mmlnone }{\prime }}}\limits _{J}\sum _{j\notin J}\int _{\Omega }|\overline{L_{j}}(\rho g_{\eta })-\alpha _{\eta }(\overline{L_{j}})|^{2}|(X_{\eta }-\overline{X_{\eta }})^{k}(N_{\delta ,q}u)_{J}|^{2}\;\;\;\;\;\;\;\;\;\;\;\; \\+C_{\eta }\,(\Vert \overline{\partial }N_{\delta ,q}u\Vert _{k-1}^{2}+\Vert \overline{\partial }^{*}N_{\delta ,q}u\Vert _{k-1}^{2}+\Vert N_{\delta ,q}u\Vert _{k-1}\Vert N_{\delta ,q}u\Vert _{k}+\Vert u\Vert _{k-2}^{2})\;.\) 
We only have to sum over \(j\notin J\)  because for \(j\in J\) , the term \(\overline{\omega _{j}}\wedge \overline{\omega _{J}}\)  vanishes. The term \(\Vert u\Vert _{k-2}^{2}\)  again results from the compactly supported term that arises from letting \((X_{\eta }-\overline{X_{\eta }})\)  act in special boundary charts; as above, the estimate follows from interior elliptic regularity of \(\Box _{\delta ,q}\) , with constants that are uniform in \(\delta \) . The order \((k-1)\)  error terms are acceptable by the induction hypothesis, and using (REF ) for the last term shows that it can be split into an absorbable term and a lower order term. From Lemma REF , we similarly obtain
\(\Vert [\overline{\partial }^{*},(X_{\eta }-\overline{X_{\eta }})^{k}]N_{\delta ,q}u\Vert ^{2} \\\lesssim \mathop {\mmlmultiscripts{\sum {\mmlnone }{\prime }}}\limits _{J}\sum _{j\in J}\int _{\Omega }|L_{j}(\rho g_{\eta }-\alpha _{\eta }(L_{j})|^{2}|(X_{\eta }-\overline{X_{\eta }})^{k}(N_{\delta ,q}u)_{J}|^{2}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;+ C_{\eta }\,\left(\Vert \overline{\partial }N_{\delta ,q}u\Vert _{k-1}^{2}+\Vert \overline{\partial }^{*}N_{\delta ,q}u\Vert _{k-1}^{2}+\Vert N_{\delta ,q}u\Vert _{k-1}\Vert N_{\delta ,q}u\Vert _{k}+ \Vert (N_{\delta ,q}u)_{N}\Vert _{k}^{2}+\Vert u\Vert _{k-2}^{2}\right)\;.\) 
The first three error terms are the same as in (REF ) and so are acceptable, as is the last. For fourth one, we use the subelliptic estimate \(\Vert (N_{\delta ,q})_{N}\Vert _{k}^{2} \lesssim \Vert \overline{\partial }N_{\delta ,q}u\Vert _{k-1}^{2}+\Vert \overline{\partial }^{*}N_{\delta ,q}u\Vert _{k-1}^{2}+\Vert N_{\delta ,q}u\Vert _{k-1}^{2}\)  ([2]}, estimate (2.96)). Using (REF ) again shows that this term is also acceptable.
For the commutator with \(\nabla \)  in (REF ), we note that it is of order \(k\) , so that the term is estimated by \(\delta C_{\eta }\Vert N_{\delta ,q}u\Vert _{k}^{2}\lesssim \delta C_{\eta }(\Vert \overline{\partial }N_{\delta ,q}u\Vert _{k,2h_{\eta }}^{2}+\Vert \overline{\partial }^{*}N_{\delta ,q}u\Vert _{k,2h_{\eta }}^{2})\)  (again by (REF ), the weight in the norms just changes the constant \(C_{\eta }\) ). If we choose \(\delta _{0}(\eta )\)  small enough, this term can therefore be absorbed when \(\delta \le \delta _{0}(\eta )\) . We are therefore left with estimating only the main terms on the right hand sides of (REF ) and (REF ), respectively.
For \(j<n\) , the contributions coming from \(\overline{L_{j}}(\rho g_{\eta })\)  and \(L_{j}(\rho g_{\eta })\) , respectively, are \(\mathcal {O}_{\eta }(\Vert \rho (X_{\eta }-\overline{X_{\eta }})^{k} N_{\delta ,q}u\Vert _{k}\)  and are again acceptable (\(\rho \)  is a subelliptic multiplier and (REF )). Because \(\alpha _{\eta }\)  is real, \(|\alpha _{\eta }(\overline{L_{j}})|^{2} = |\alpha _{\eta }(L_{j})|^{2}\) , and we only have to estimate \(\int _{\Omega }|\alpha _{\eta }(L_{j})|^{2}|(X_{\eta }-\overline{X_{\eta }})^{k}(N_{\delta ,q}u)_{J}|^{2}=\int _{\Omega }|\alpha _{\eta }((X_{\eta }-\overline{X_{\eta }})^{k}(N_{\delta ,q}u)_{J}L_{j})|^{2}\) . We do this via (REF ) in Proposition REF , applied to the \((0,1)\) –form \((X_{\eta }-\overline{X_{\eta }})^{k}(N_{\delta ,q}u)_{J}\overline{\omega _{j}}\) .Then \(L_{u}=((X_{\eta }-\overline{X_{\eta }})^{k}(N_{\delta ,q}u)_{J}L_{j}\) , and the resulting estimate is
\(\int _{\Omega }|\alpha _{\eta }(L_{j})|^{2}|(X_{\eta }-\overline{X_{\eta }})^{k}(N_{\delta ,q}u)_{J}|^{2} \\\lesssim (1-\eta )\left(\Vert \overline{\partial }((X_{\eta }-\overline{X_{\eta }})^{k}(N_{\delta ,q}u)_{J}\overline{\omega _{j}})\Vert ^{2}+\Vert \overline{\partial }^{*}((X_{\eta }-\overline{X_{\eta }})^{k}(N_{\delta ,q}u)_{J}\overline{\omega _{j}})\Vert ^{2}\right) \\+ C_{\eta }\left(\Vert \overline{\partial }((X_{\eta }-\overline{X_{\eta }})^{k}(N_{\delta ,q}u)_{J}\overline{\omega _{j}})\Vert _{\Omega _{\eta }}^{2}+\Vert \overline{\partial }^{*}((X_{\eta }-\overline{X_{\eta }})^{k}(N_{\delta ,q}u)_{J}\overline{\omega _{j}})\Vert _{\Omega _{\eta }}^{2}\right) \\+C_{\eta }\Vert (X_{\eta }-\overline{X_{\eta }})^{k}N_{\delta ,q}u\Vert _{-1}^{2}\;.\) 
The error terms are acceptable; this is clear for the \(\Vert \cdot \Vert _{-1}\)  term (also using (REF )). For the others, it follows from interior elliptic regularity of \(\Box _{\delta ,q}\) , with constants that are uniform in \(\delta \) . The \(\overline{\partial }\)  and \(\overline{\partial }^{*}\)  terms in the second line of (REF ) are dominated (uniformly in \(\eta \)  and \(\delta \) ) by
\(\sum _{s=1}^{n}\Vert \overline{L_{s}}((X_{\eta }-\overline{X_{\eta }})^{k}(N_{\delta ,q}u)_{J})\Vert ^{2}+\Vert L_{j}((X_{\eta }-\overline{X_{\eta }})^{k}(N_{\delta ,q}u)_{J})\Vert ^{2}+\Vert (X_{\eta }-\overline{X_{\eta }})^{k}N_{\delta ,q}u\Vert ^{2} \\\lesssim \Vert \overline{\partial }((X_{\eta }-\overline{X_{\eta }})^{k}N_{\delta ,q}u)\Vert ^{2}+\Vert \overline{\partial }^{*}((X_{\eta }-\overline{X_{\eta }})^{k}N_{\delta ,q}u)\Vert ^{2}\;;\) 
we have used the basic estimate for the barred derivatives for the first term, maximal estimates for the second (see (REF ); note again that \(q_{0}\le q\) , and Lemma REF ), and the basic \(L^{2}\)  estimate for the third.
There remain the terms in (REF ) and (REF ) where \(j=n\) . In (REF ), we have that \(|\overline{L_{n}}(\rho g_{\eta })-\alpha _{\eta }(\overline{L_{n}})|^{2}=0\)  on the boundary, due to the choice of \(g_{\eta }\) . The contribution from this term is thus \(\mathcal {O}_{\eta }(\Vert \rho (X_{\eta }-\overline{X_{\eta }})^{k}N_{\delta ,q}u\Vert ^{2})\) . Again because \(\rho \)  is a subelliptic multiplier and (REF ), this term is acceptable. In (REF ), the term with \(j=n\)  occurs only in a sum over \(J\)  with \(n\in J\) . That is, \((N_{\delta ,q}u)_{J}\)  is a component of the normal part \((N_{\delta ,q}u)_{N}\)  of \(N_{\delta ,q}u\)  and so is subelliptic; we conclude as above that it is acceptable.
Combining (REF )–(REF ), and the remark above about the commutator with \(\nabla \)  in (REF ) gives
\(\Vert [\overline{\partial },(X_{\eta }-\overline{X_{\eta }})^{k}]N_{\delta ,q}u\Vert ^{2}+\Vert [\overline{\partial }^{*},(X_{\eta }-\overline{X_{\eta }})^{k}]N_{\delta ,q}u\Vert ^{2}+ \delta \Vert [\nabla ,(X_{\eta }-\overline{X_{\eta }})^{k}]N_{\delta ,q}u\Vert ^{2} \\\lesssim (1-\eta )\left(\Vert \overline{\partial }((X_{\eta }-\overline{X_{\eta }})^{k}N_{\delta ,q}u)\Vert ^{2} + \Vert \overline{\partial }^{*}((X_{\eta }-\overline{X_{\eta }})^{k}N_{\delta ,q}u)\Vert ^{2}\right) \\+ terms\; acceptable \;terms \\\lesssim (1-\eta )\left( \Vert [\overline{\partial },(X_{\eta }-\overline{X_{\eta }})^{k}]N_{\delta ,q}u\Vert ^{2}+\Vert [\overline{\partial }^{*},(X_{\eta }-\overline{X_{\eta }})^{k}]N_{\delta ,q}u\Vert ^{2} \right) \\+ (1-\eta )\left(\Vert (X_{\eta }-\overline{X_{\eta }})^{k}\overline{\partial }N_{\delta ,q}u\Vert ^{2} + \Vert (X_{\eta }-\overline{X_{\eta }})^{k}\overline{\partial }^{*}N_{\delta ,q}u\Vert ^{2}\right) \\+ acceptable \;terms\;.\) 
For \((1-\eta )\)  small enough, the fourth line in (REF ) can be absorbed into the first line, giving
\(\Vert [\overline{\partial },(X_{\eta }-\overline{X_{\eta }})^{k}]N_{\delta ,q}u\Vert ^{2}+\Vert [\overline{\partial }^{*},(X_{\eta }-\overline{X_{\eta }})^{k}]N_{\delta ,q}u\Vert ^{2}+ \delta \Vert [\nabla ,(X_{\eta }-\overline{X_{\eta }})^{k}]N_{\delta ,q}u\Vert ^{2} \\\lesssim (1-\eta )\left(\Vert (X_{\eta }-\overline{X_{\eta }})^{k}\overline{\partial }N_{\delta ,q}u\Vert ^{2} + \Vert (X_{\eta }-\overline{X_{\eta }})^{k}\overline{\partial }^{*}N_{\delta ,q}u\Vert ^{2}\right)+ acceptable \;terms\;.\) 
The left hand side of (REF ) is what is needed to estimate the sum of the left hand sides of (REF ), (REF ), and (REF ). Thus
\(\Vert \overline{\partial }N_{\delta ,q}u\Vert _{k,2h_{\eta }}^{2}+\Vert \overline{\partial }^{*}N_{\delta ,q}u\Vert _{k,2h_{\eta }}^{2}+\delta \Vert \nabla N_{\delta ,q}u\Vert _{k,2h_{\eta }}^{2} \\\lesssim (1-\eta )\left(\Vert (X_{\eta }-\overline{X_{\eta }})^{k}\overline{\partial }N_{\delta ,q}u\Vert ^{2} + \Vert (X_{\eta }-\overline{X_{\eta }})^{k}\overline{\partial }^{*}N_{\delta ,q}u\Vert ^{2}\right)+ acceptable \;terms\;,\) 
where `acceptable terms' stands for terms that are of order at most \((k-1)\)  in \(\overline{\partial }N_{\delta ,q}u\) , \(\overline{\partial }^{*}N_{\delta ,q}u\) , or \(\delta ^{1/2}\nabla N_{\delta ,q}u\) , or are the same as the terms on the left hand side of (REF ), but with a small constant in front (for \((1-\eta )\)  small enough and then \(\delta \le \delta _{0}(\eta )\) ), or are dominated by \(\Vert u\Vert _{k}^{2}\) . After combining the estimates \(\Vert (X_{\eta }-\overline{X_{\eta }})^{k}\overline{\partial }N_{\delta ,q}u\Vert ^{2}\lesssim \Vert \overline{\partial }N_{\delta ,q}u\Vert _{k,2h_{\eta }}^{2}+C_{\eta }\Vert N_{\delta ,q}u\Vert _{k-1}^{2}\)  and \(\Vert (X_{\eta }-\overline{X_{\eta }})^{k}\overline{\partial }^{*}N_{\delta ,q}u\Vert ^{2}\lesssim \Vert \overline{\partial }^{*}N_{\delta ,q}u\Vert _{k,2h_{\eta }}^{2}+C_{\eta }\Vert N_{\delta ,q}u\Vert _{k-1}^{2}\)  with the induction hypothesis, and absorbing terms, we obtain that there are a constant \(C\) , \(\eta \in (0,1)\) , and \(\delta _{0}\)  such that
\(\Vert \overline{\partial }N_{\delta ,q}u\Vert _{k,2h_{\eta }}^{2}+\Vert \overline{\partial }^{*}N_{\delta ,q}u\Vert _{k,2h_{\eta }}^{2}+\delta \Vert \nabla N_{\delta ,q}u\Vert _{k,2h_{\eta }}^{2} \le C\Vert u\Vert _{k}^{2}\;,\;0<\delta \le \delta _{0}\;.\) 
This completes the induction on \(k\) ; (REF ) holds for all \(k\in \mathbb {N}\) , with \(C\) , \(\eta \) , and \(\delta _{0}\)  depending on \(k\) .
Fix \(k\in \mathbb {N}\) . In view of (REF ), and choosing \(\eta \)  so that (REF ) holds, we see that there is \(\delta _{0}(k)>0\)  and a constant \(C_{k}\)  such that
\(\Vert N_{\delta ,q}u\Vert _{k}^{2} \le C_{k}\Vert u\Vert _{k}^{2}\;,\,0<\delta \le \delta _{0}(k)\;.\) 
(The weighted and unweighted norms are equivalent, with constants depending on \(\eta \) ; once \(\eta \)  is fixed, this dependence is no longer relevant.) As we said at the beginning of this section, letting \(\delta \rightarrow 0\)  transfers this estimate to \(N_{q}\) . Since \(k\)  was arbitrary, so we have now shown that if \(N_{q+1}\)  satisfies Sobolev estimates for \(q\ge q_{0}\) , then so does \(N_{q}\) . This concludes the downward induction on the degree \(q\) , and completes the proof of Theorem REF .
